Ohio Assessments for Educators (OAE) Mathematics Practice Exam

The derivative of the natural logarithm function, ln(x), is derived from the definition of the derivative and the properties of logarithmic functions. The rule states that if you have a function in the form of ln(u), where u is a function of x, the derivative is given by the chain rule as (1/u) * (du/dx). When u is simply x, the expression simplifies to 1/x.

Therefore, the derivative of ln(x) is 1/x. This reflects the relationship between the rate of change of the ln function at any point x and the value of x itself. As x increases, the value of the derivative decreases but remains positive for x > 0, indicating that the function ln(x) is increasing but at a decreasing rate.

In contrast, the other options represent different mathematical expressions that do not describe the derivative of ln(x). The expression 1/x² signifies a function that decreases more rapidly than 1/x. The expression ln(x)/x combines both logarithmic and linear components but does not accurately represent the derivative. Lastly, x² implies a completely different growth behavior, which is relevant to polynomial functions rather than logarithmic functions.